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Theorem opth 4255
Description: The ordered pair theorem. If two ordered pairs are equal, their first elements are equal and their second elements are equal. Exercise 6 of [TakeutiZaring] p. 16. Note that 𝐶 and 𝐷 are not required to be sets due our specific ordered pair definition. (Contributed by NM, 28-May-1995.)
Hypotheses
Ref Expression
opth1.1 𝐴 ∈ V
opth1.2 𝐵 ∈ V
Assertion
Ref Expression
opth (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem opth
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 opth1.1 . . . 4 𝐴 ∈ V
2 opth1.2 . . . 4 𝐵 ∈ V
31, 2opth1 4254 . . 3 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶)
41, 2opi1 4250 . . . . . . 7 {𝐴} ∈ ⟨𝐴, 𝐵
5 id 19 . . . . . . 7 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
64, 5eleqtrid 2278 . . . . . 6 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴} ∈ ⟨𝐶, 𝐷⟩)
7 oprcl 3817 . . . . . 6 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
86, 7syl 14 . . . . 5 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
98simprd 114 . . . 4 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐷 ∈ V)
103opeq1d 3799 . . . . . . . 8 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐵⟩)
1110, 5eqtr3d 2224 . . . . . . 7 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
128simpld 112 . . . . . . . 8 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐶 ∈ V)
13 dfopg 3791 . . . . . . . 8 ((𝐶 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐶, 𝐵⟩ = {{𝐶}, {𝐶, 𝐵}})
1412, 2, 13sylancl 413 . . . . . . 7 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐵⟩ = {{𝐶}, {𝐶, 𝐵}})
1511, 14eqtr3d 2224 . . . . . 6 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐵}})
16 dfopg 3791 . . . . . . 7 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
178, 16syl 14 . . . . . 6 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
1815, 17eqtr3d 2224 . . . . 5 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {{𝐶}, {𝐶, 𝐵}} = {{𝐶}, {𝐶, 𝐷}})
19 prexg 4229 . . . . . . 7 ((𝐶 ∈ V ∧ 𝐵 ∈ V) → {𝐶, 𝐵} ∈ V)
2012, 2, 19sylancl 413 . . . . . 6 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐶, 𝐵} ∈ V)
21 prexg 4229 . . . . . . 7 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → {𝐶, 𝐷} ∈ V)
228, 21syl 14 . . . . . 6 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐶, 𝐷} ∈ V)
23 preqr2g 3782 . . . . . 6 (({𝐶, 𝐵} ∈ V ∧ {𝐶, 𝐷} ∈ V) → ({{𝐶}, {𝐶, 𝐵}} = {{𝐶}, {𝐶, 𝐷}} → {𝐶, 𝐵} = {𝐶, 𝐷}))
2420, 22, 23syl2anc 411 . . . . 5 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ({{𝐶}, {𝐶, 𝐵}} = {{𝐶}, {𝐶, 𝐷}} → {𝐶, 𝐵} = {𝐶, 𝐷}))
2518, 24mpd 13 . . . 4 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐶, 𝐵} = {𝐶, 𝐷})
26 preq2 3685 . . . . . . 7 (𝑥 = 𝐷 → {𝐶, 𝑥} = {𝐶, 𝐷})
2726eqeq2d 2201 . . . . . 6 (𝑥 = 𝐷 → ({𝐶, 𝐵} = {𝐶, 𝑥} ↔ {𝐶, 𝐵} = {𝐶, 𝐷}))
28 eqeq2 2199 . . . . . 6 (𝑥 = 𝐷 → (𝐵 = 𝑥𝐵 = 𝐷))
2927, 28imbi12d 234 . . . . 5 (𝑥 = 𝐷 → (({𝐶, 𝐵} = {𝐶, 𝑥} → 𝐵 = 𝑥) ↔ ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)))
30 vex 2755 . . . . . 6 𝑥 ∈ V
312, 30preqr2 3784 . . . . 5 ({𝐶, 𝐵} = {𝐶, 𝑥} → 𝐵 = 𝑥)
3229, 31vtoclg 2812 . . . 4 (𝐷 ∈ V → ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷))
339, 25, 32sylc 62 . . 3 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐵 = 𝐷)
343, 33jca 306 . 2 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (𝐴 = 𝐶𝐵 = 𝐷))
35 opeq12 3795 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
3634, 35impbii 126 1 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2160  Vcvv 2752  {csn 3607  {cpr 3608  cop 3610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616
This theorem is referenced by:  opthg  4256  otth2  4259  copsexg  4262  copsex4g  4265  opcom  4268  moop2  4269  opelopabsbALT  4277  opelopabsb  4278  ralxpf  4791  rexxpf  4792  cnvcnvsn  5123  funopg  5269  funinsn  5284  brabvv  5942  xpdom2  6857  xpf1o  6872  djuf1olem  7082  enq0ref  7462  enq0tr  7463  mulnnnq0  7479  eqresr  7865  cnref1o  9680  fisumcom2  11478  fprodcom2fi  11666  qredeu  12129  fnpr2ob  12816
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